L9a2

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_18,12,5,11 X_10,18,11,17 X_2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -2 u 4 -2 v u 3 + 2 u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+3 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 - z 5 a -1 + 5 z 5 a -3 - z 5 a -5 -3 z 3 a -1 + 7 z 3 a -3 -3 z 3 a -5 + z a -3 - z a -5 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 8 a -2 -2 z 8 a -4 -3 z 7 a -1 -7 z 7 a -3 -4 z 7 a -5 + 5 z 6 a -2 + 2 z 6 a -4 -4 z 6 a -6 - z 6 + 12 z 5 a -1 + 24 z 5 a -3 + 8 z 5 a -5 -4 z 5 a -7 + 3 z 4 a -4 + 3 z 4 a -6 -3 z 4 a -8 + 3 z 4 -12 z 3 a -1 -22 z 3 a -3 -6 z 3 a -5 + 3 z 3 a -7 - z 3 a -9 - z 2 a -2 + 2 z 2 a -6 + 2 z 2 a -8 - z 2 + z a -1 + 3 z a -3 + 2 z a -5 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

3 is the signature of L9a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$